This is what I would do- check to make sure the ground cable where it attaches to the body is clean and well connected. No grease inbetween the connectors.
I would double check the positive connector to the battery and make sure it is clean and then do the same for the negative.
>>>Always disconnect and connect the ground wire first on batteries. <<<
If the cable connector to the battery is hard to get on the battery terminal, try inserting a large screw driver into the space by the battery connector bolt and see if you can spread the ends of the connector. Be sure to have the nut and bolt unscrewed enough to allow this.
Once all the above is done I would check all of the fues. Rotate the fuses in their sockets. If they are aluminum they often corrorde and often it is hard to see the corrosion.
If would turn the car on and then off again. I would lock the doors and then open them again. This is under the presumption that it is a factory alarm.
If the problem continues I would also check the battery for water and for a proper charge. I would also check the alternator belt to make sure the battery is getting enough charge.
Keep us posted,
'03 E320 Wagon-Sold
'95 E320 Wagon-Went to Ex
'93 190E 2.6-Wrecked
'91 300E-Went to Ex
'65 911 Coupe (#302580)